'Handel's Temperament' revisited
“Onemust take refuge in temperament when tuning claviers and harps. Many books make as much to do of this as if the welfare of the entire world depended on a single clavier.”
Johann Mattheson 
Posted: 8 August 2016
Revised: 16 August 2016
Copyright © C E Pykett 2016
Abstract. In about 1780 a description was published in London of what has become known as 'Handel's Tuning' or 'Handel's Temperament' Although the association with Handel has little foundation, this article suggests that it is worth revisiting the temperament attributed to him for several other reasons. One is the unfortunate fact that many modern realisations of it are wrong because they are incompatible with the complete set of tuning instructions given. These illuminate tuning practices at a time when beat counting was uncommon and, while therefore vague on detail, they nevertheless incorporate an explicit set of overarching constraints which place definite bounds on the possible outcomes. In particular, all fifths must be flat rather than pure or sharpened, and all thirds must be considerably sharp. It is therefore regrettable that the modern realisations of this temperament which are embodied in electronic tuning devices are mostly wrong because they incorporate pure fifths. It is also impossible to conclude, as some authors have done, that the temperament was just another variation on the meantone tunings which were common at that time in Britain. On the contrary, it is difficult to see how the temperament can be other than a mildly unequal one. Consequently it probably lies on the haphazard path which ultimately led to equal temperament in the nineteenth century rather than being merely another example of an already-outmoded meantone approach, although the temperament is not equal because the tuning instructions also dictate that the fifths should differ in their deviations from pure.
The frustratingly unfocused tuning instructions nevertheless allow for several subtle variations in realising a version of the temperament, and therefore I took advantage of this to produce my own which is described in the article. It addresses the tuning instructions at a level of detail not found elsewhere, providing a reading which I therefore believe to be novel. The outcome is a pleasing well temperament with a range of subtle key colours which can render music in any key. Because of the imprecision of the tuning instructions it is not possible to interpret them in a single, unequivocal fashion - the best that can be done is to come up with one of several possible variations on the theme of 'mildly unequal'. That described here is represented as a set of deviations from equal temperament for each note in the scale, a format which is well suited to most of today's electronic tuning devices and apps, thus it should be reasonably simple to evaluate in practice.
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In about 1780 a description was published in London, England of what has become known as
'Handel's Tuning' or 'Handel's Temperament'
. It is a means of tuning (a method for setting a temperament) on instruments such as the organ and harpsichord, and it was said to have been invented, or at least used, by Handel himself. The association with Handel is probably spurious
(, ) and as far as this article is concerned it is best ignored, to be pigeonholed alongside similar claims elsewhere for the temperaments allegedly employed by J S Bach. Such unfalsifiable hypotheses, with their
inherent encouragement of eternal dispute, do not find a natural home with most scholars.
Disregarding major errors such as these, the somewhat vague tuning instructions nevertheless allow for many subtle variations in realising a version of the temperament, and therefore I took advantage of this to produce my own which is now described here. It was developed independently of Johnson's research since I was unable to read her paper until after I had completed my version. Moreover, I am unaware of any other attempt to address and interpret the tuning instructions at the level of detail to be found in this article since Jorgensen published his somewhat perplexing deductions in 1991 , and I believe this to be novel.
The tuning instructions from 1780
 are entirely non-numerical and qualitative in the sense that beat counting is not used, indeed beats or anything remotely resembling them are not mentioned at all. Instead there are vague references to interval 'quality' or purity such as
"let the Fifth be nearer perfect than the last tho' not quite". It is therefore next to impossible to produce a unique, unequivocal version of this temperament because different tuners will arrive at different results, and the same tuner will have difficulty reproducing identical results on each occasion if tuning by ear alone.
This does not merely reflect some nonchalant musing on my part because it is difficult to see how the outcome can be other than mildly unequal. Although the tuning instructions are vague at a detailed level, they nevertheless lay down a number of unequivocal overarching constraints which have been summarised above, and together these limit one's room for manoeuvre when one tries to implement the temperament for real. Albeit somewhat disguised at first sight, the instructions simply tune the twelve fifths and fourths in a single octave in a more or less conventional manner, and the fifths must all be flat, not pure or sharp. Therefore one has to be careful not to flatten any of them too much otherwise it will be impossible to compensate elsewhere to prevent the sum of the deviations from pure exceeding the Pythagorean Comma. This problem does not affect strongly unequal temperaments because any number of fifths can be made pure or sharpened to compensate for the flattened ones. Taking an extreme example, in sixth-comma meantone (the so-called 'Silbermann' temperament) this leads to the single grossly sharpened Wolf interval. But because the quite explicit constraints of the 'Handel' temperament rule this out, it is difficult to understand how anyone could suggest that it resembles a meantone tuning, as Johnson did in reference  for instance. I found that trying to tune the temperament was a salutary, if frustrating and time-consuming, exercise in ferreting out what the instructions were actually saying, rather than making baseless assumptions as to what one might like them to say.
Given this, one can go further. Although variants of meantone tuning were common in England during the 18th century and they persisted into the 19th, there was also some pressure to adopt less aggressive tunings which might have been imported from continental Europe, and eventually they led to the comparatively late adoption of equal temperament in Britain. It is therefore my contention that the so-called 'Handel' temperament was an evolutionary step along this newer road, rather than yet another example of an already-outmoded meantone approach.
The temperament was set up using a digital organ for convenience, using a mild-toned 8 foot Principal stop not unlike a Dulciana. Such stops would have been common in England in the 18th century, and I considered it important to use one whose harmonic spectrum would have been similar to those used to tune an organ to this temperament. Tuning, and therefore setting any temperament, is a matter of detecting beats. Whether one counts or times them or not is largely immaterial - the fact remains that they exist except when an interval reaches purity and this is what a tuner listens for, consciously or otherwise, when tuning pure or setting a deviation from pure. It follows that the harmonic spectrum of the stop used for setting a temperament is important because the beats arise exclusively from harmonics above the fundamental frequency except when tuning unisons and octaves. For example, the strongest and lowest-frequency beat of a fifth arises between the third harmonic of the lower note and the second harmonic of the upper. Other beats at integer multiples of this frequency also exist if there are sufficient audible harmonics, but they are of lower amplitudes because the generating harmonics themselves become weaker in flue pipe spectra as their frequencies increase. In the case of a fourth the predominant beat arises from the fourth harmonic of the lower note and the third of the upper, and for a major third it involves the fifth harmonic of the lower note and the fourth of the upper. Therefore the stop chosen for tuning purposes must allow these harmonics to be heard, yet there should not be too many others which might confuse the ear if too many higher-order beats are generated. For similar reasons one should desirably choose a stop whose timbre (harmonic spectrum) relates to the epoch in which the temperament of interest arose because the temperament designer would have had in mind the tone colour, and thus the beat-generating properties, of the pipes which would be used to tune the instrument. Hence my choice of a mild Principal tone.
The tuning instructions were presented in the original document as a set of eight major triads written as broken chords together with various instances of octaves (which are of course tuned pure). Each triad has its own set of instructions and we shall now consider them in turn. They were printed using the Georgian 'long s' typeface which looked similar to an 'f' but here the modern letterform has been substituted. The tuning for each note in the octave which is derived from the instructions is represented here as a deviation in cents from its ET counterpart. This was thought to be the most convenient format for today's tuners, who will most likely wish to use the offsets in some form of electronic tuning device. Because of the vagueness of the instructions however, the data are represented to a precision of only one cent with no decimal places, as a greater precision was not warranted.
First chord (C major):
The original instructions state:
"In this Chord tune the Fifth pretty flat and the Third considerably too sharp. NB. the Fifth will not bear to be reduced so much below its true accord, as the Third will to be raised above it"
One interpretation of these instructions is as follows:
Second chord (G major):
"Let the Fifth be nearer perfect than the last tho' not quite, tune the Third a Fifth to E, make it good but just bearing flat".
This implies that the interval G-D should be given the merest nudge flat from pure. Tuning D one cent sharp relative to its ET value results in a beat frequency of 0.66 Hz flat from pure for this interval, or just over one beat every two seconds. In ET the beat would be twice as fast at 1.34 Hz. Therefore the instructions appear to have been satisfied in a qualitative sense. (Note that these numerical values for beat frequencies assume that middle C was first tuned to 261.63 Hz as stated above. Other choices will give different results).
Tune the octaves:
"Tune all Octaves perfect".
This instruction does not call for further discussion beyond observing that it tunes three notes which are used in subsequent steps.
Third chord (D major):
"Tune A, a good fifth to D, trying it at the same time with E, above already tuned. Tune the Third a fifth to B, and let it be near as flat as the fifth in the 1st Chord, this will in some measure bring down the sharpness of the Third".
Middle D was tuned from its octave in the previous step, and using it the interval D-A should then be tuned as a "good fifth". As with so much else in these instructions it is not at all clear what is meant by the adjective "good". For example, all fifths in ET are tuned flat by almost
two cents - would these have been considered "good" in the 18th century? We know that the interval cannot be pure because we have noted several times already that perfectly tuned fifths are disallowed in this temperament. Therefore, as with beauty, 'goodness' must depend as much on the ears of the beholder as anything else. Here I sharpened A by one cent above its ET value to bring its interval with D closer to a pure fifth. (Note that this means the frequency of A will no longer be exactly 440 Hz given the frequency already chosen for C. However it can be easily adjusted back to this value when the entire temperament has been set up, and this will be described later). This choice endows the interval D-A with a beat frequency of 1 Hz, hopefully making it indeed something which could be considered a "good fifth".
Fourth chord (A Major):
"Tune the Third a fifth to F, sharp [sic - presumably this means 'F sharp'] already tuned, let its bearing be the same as the Third in the last Chord".
On the face of it this implies that the interval F#-C# should be narrowed by one cent from ET in the same way that B-F# was narrowed in the previous step. Therefore C# is tuned with an offset of -1 cent from ET.
"Tune the Third very fine rather bearing sharp, for in fact all Thirds must be tuned sharp more or less, as all fifths should be flat".
This instruction, relating to G# because E and B have been tuned already, is perhaps the most obscure of all. It certainly gives the most leeway for tuners to impose their signature on their version of the temperament. At one extreme it could be taken to imply that the interval E-G# should be tuned just sharp of pure, which implies an offset for G# from ET of anything up to a gross -13 cents or so. After some experiments I concluded that anything beyond an offset of
-6 cents made little sense for a well temperament, being too extreme for my ears. Eventually I settled on a range of acceptable offsets lying between -1 and -4 cents. The value adopted here is -1 cent, in keeping with the presumed well-tempered ethos of the temperament as a whole.
(This chord was thrown onto a C-clef in the original document but the F-clef has been used here as it is probably more familiar).
"Tune the A, an Octave to A above, when you have drawn the F, to a perfect fifth with C, give it a little inclination higher, that being the same as if you had tuned your C, a fifth to F, giving the C a little flatness".
This instruction has given rise to misunderstandings on the part of some authors. For instance, Di Veroli correctly noted that all fifths should be flat but then read this as implying that F should be tuned pure to C . In fact there is no such incompatibility among the tuning instructions because the obvious intention here is that the interval should first be drawn into perfect tune before nudging it slightly flat, an eminently reasonable suggestion clearly aimed at the practical tuner. He also concluded that the temperament as a whole was a "mildly unequal" one "which fits Handel's music quite well", and that it was "far away from the usual tuning practices described in some sources dating from Handel's lifetime". These deductions chime with my own as discussed earlier.
Thus tenor A (the A below middle C) is first tuned to its octave which has been tuned already, as has middle C. Then, this
admittedly rather confusing instruction is simply saying that the interval tenor F-middle C should be narrowed slightly from pure in the same way as all the others. I gave F an offset of -1 cent relative to ET, making the interval slightly wider and closer to pure than in ET.
"Tune the fifth as in the 6th Chord".
B flat was given an offset of -2 cents from ET. This makes the interval with F the same as in the sixth chord (slightly wider and closer to pure than in ET), which is what the instructions call for.
"Tune the fifth as in the last chord".
The only note to be tuned here is E flat, to which I gave an offset of -3 cents. Therefore the interval with B flat became the same as in the previous case for the seventh chord (slightly wider and closer to pure than in ET), which is what the instructions call for.
The frequency offsets in cents from equal temperament for each note which resulted from the tuning procedure above are summarised in the tables below for convenience. These can be inserted directly into most electronic tuning devices or apps. Two sets of data are given. The first (Table 1) simply pulls together the numbers from the foregoing section. It will be recalled that these were referenced to middle C with a frequency of 261.63 Hz. Since A was given an offset of +1 cent, its frequency is 440.25 Hz.
Table 1 . Note offsets from equal temperament in cents (middle C = 261.63 Hz; A = 440.25 Hz)
However it might be preferred to reference the data to A = 440.00 Hz. As this note has an offset of +1 cent in Table 1 this now has to be removed, and this means that all other offsets have to be reduced by the same amount. This is done in Table 2. The frequency of middle C is reduced accordingly.
Table 2 . Note offsets from equal temperament in cents (middle C = 261.48 Hz; A = 440.00 Hz)
Having said this, it is not really necessary to represent the relative tunings of the fifths in circular form on the page as a tabular representation is just as convenient if not more so, and this is given below in Table 3. Intervals are expressed here using the tuner's convention of referring to notes either as naturals or sharps, which means that not all of them make sense musically. In particular the interval A# to F is not, musically speaking, a fifth even though it brackets the same number of physical semitones on a modern keyboard as all the others (it should really be written as A# - E#). However it is presumed readers will understand this.
In this temperament all fifths are tuned flat from pure because the instructions specify that this must be so. Therefore there are no pure or sharpened fifths as there are in Jorgensen's version  or as implied by Johnson . In equal temperament all the deviations would be the same and equal to one twelfth of the Pythagorean Comma or -1.955 cents . Because the deviations in the second column of the table are not the same, this indicates that the temperament is an unequal one. However three of the intervals nevertheless take the ET value whereas five are smaller (-0.955 cents), which means these intervals are nearer to pure than in ET. Three have larger deviations from pure than in ET at -2.955 cents and one is larger still at -3.955. These variations are reflected in the beat frequencies one hears when the two notes bracketing the interval are sounded together, and they are tabulated in the right hand column . The three ET intervals beat at or close to 1 Hz, those with smaller deviations beat more slowly and those with larger ones more quickly. (Note that the beat frequencies depend on the pitch of the stop and the octave in which the intervals are played whereas the cent values are independent of frequency. Here the numbers relate to an 8 foot stop in the octave starting at middle C). The slowest beat at 0.5 Hz occurs for the interval D# - A#, which is one beat every two seconds, making this interval nearly perfect. The other intervals with the smallest deviation values are also very close to pure (within one cent). The fastest beat at nearly 3 Hz, three beats per second, occurs with G# - D#, and this is the nearest one gets to the 'Wolf' intervals which occur in some other unequal temperaments. Even so, this interval is far removed from the unpleasantness of any interval meriting the Wolf soubriquet. The variations in beat frequencies of these and other intervals are responsible for the impression of key colour which one receives from this temperament about which more will be said presently.
Another column of the table represents the deviations from pure as fractions of the Pythagorean Comma. For the three ET intervals these values are all 1/12, whereas for the others the fractions are larger or smaller. It is noteworthy that the smallest fraction is 1/24. A divisor of 24 does not appear frequently in unequal temperaments from this era, and particularly so when one goes further back in time to the 17th and early 18th centuries. It only seems to make an appearance later in the 18th century in temperaments such as certain versions of the French Ordinaire in which the fraction 5/24 occurs. Two other examples, mentioned by Padgham , occur in the Finchcocks and Oakes Park English organs where fractions of 1/24, 7/24 and 11/24 arise (Finchcocks) and 1/24, 5/24, 7/24 and 9/24 (Oakes Park). The English examples might be no more than interesting coincidences because the temperaments could only be reconstructed from fragmentary data, and Padgham's book cannot always be relied upon in any case in terms of the numbers it contains . Nevertheless the coincidence is worthy of note all the same, particularly in the context of the evolution of English temperament. The minute deviation of only 1/24th of the Pythagorean Comma in some of these temperaments (including the one derived here) implies that tuners were honing their art towards greater precision by the mid to late 18th century because it represents a deviation of only one cent from the pure interval. It might also reflect an intermediate phase of the haphazard drift towards equal temperament which subsequently became better established in Britain as the 19th century advanced. Statements in the tuning instructions above such as "let the Fifth be nearer perfect than the last tho' not quite" and "tune the ... F, to a perfect fifth with C, give it a little inclination higher" give the impression that some exactness is required to nudge intervals only a smidgeon away from pure, even though the numerical precision of specified beat frequencies did not appear in the tuning instructions. This is some distance removed from many earlier temperaments which were strongly unequal with multiple perfect fifths on the one hand, much larger fractions of the Comma (such as 1/4th and 1/6th) for the detuned ones on the other, the inclusion of Wolf intervals and the appearance of several unusable keys. Therefore the picture of this temperament seems to be painted on a canvas which confirms my reading of it as a mildly unequal one containing hazy pre-echoes of ET.
It is next to impossible to describe the sound of something as complex as a musical temperament in words or numbers. One needs to try it for oneself. However a few remarks might be of interest. Firstly all keys are useable and it is therefore a well temperament. Secondly a range of subtle key colours is noticeable.
Thirdly it is worth remembering that temperaments produce different effects on different instruments, such as the organ compared with the harpsichord. There are several reasons for this but an important one is seldom mentioned. This reason is that the organ is unique among keyboard instruments in that its sound while it is being tuned is usually very different to that when it is used to render music. This is because tuning is done using a single stop as discussed above, whereas several quite different stops and many combinations thereof are available for performing music. Although music will sometimes be played on the stop used for tuning alone, most of the time it will not. No other keyboard instrument has this attribute. A single Principal-toned stop is usually used for tuning in which harmonics up to at least the fifth must be clearly audible, otherwise the necessary beats will not exist which are needed to inform the ear that the interval of a major third is pure or not. As explained earlier this interval only generates beats if the fourth and fifth harmonics of the two notes are audible. Yet a Flute stop either does not have such an extended harmonic retinue or, if it does, the harmonic amplitudes fall off much faster than for a Principal. If they did not, the stop would not sound like a Flute at all. Therefore one either does not perceive the fast beats of the sharpened thirds which characterise equal temperament when playing on a Flute stop, or they are insignificant. Consequently this often-deprecated property of ET is suppressed for Flutes, which can therefore sound pretty much as attractive in ET as they do in (certain keys of) some unequal temperaments where the thirds are better in tune. Hence one cannot be too prescriptive in condemning or approving temperaments for the organ, because their subjective effects vary dramatically for different stops and combinations of stops. This needs to be borne in mind because the version of 'Handel's' temperament which has emerged here, although an unequal temperament, shares with ET its rather rough major thirds - as the tuning instructions suggest that it should. The point is that one does not necessarily perceive them as 'rough' if the stop(s) being used only have weak fourth, fifth and subsequent harmonics.
However if one plays on stops with greater harmonic development such as Principals the effect of the sharpened thirds emerges, and together with the beats between some other intervals this temperament is found to have a variety of distinct key colours. The colours are not as strongly differentiated as in a more obviously unequal temperament but their presence is noticeable and subtly attractive nonetheless. Key colour is largely a property of how much the consonant intervals deviate from pure. These intervals are mainly the major and minor thirds, the fifths and their inversions (minor and major sixths and fourths respectively). The remaining intervals (semitone, whole tone, etc) are more dissonant because there are far fewer harmonics of the two notes whose frequencies nearly or completely coincide, therefore they sound harsher to the ear regardless of temperament. Consequently the subjective effect of a temperament is influenced strongly by the harmonic relationships between just two intervals - the fifths and the major thirds - because these and their inversions involve the pairs of notes which form most of the consonant intervals. In turn these manifest themselves to the ear by a complex and ever-shifting web of beats between their harmonics when music is played. The shimmer of the beats differs in different keys, creating an aural tapestry which is strongly dependent on temperament. Thus arises key colour.
Therefore we need to discuss how the fifths and thirds beat to arrive at a primitive understanding of why this temperament sounds as it does. The fifths have been covered already in the previous section and Table 3, so here we dwell briefly on the thirds.
Table 4 shows the beat frequencies of the thirds both for equal temperament and this realisation of 'Handel's' temperament in the middle octave of an 8 foot stop. There are several differences. For ET, there is a gradual increase of beat frequency as one ascends the scale whereas for the temperament realised here the values jump up and down from one interval to the next in most cases. This behaviour is most noticeable, both in terms of the numbers in the table and when listening to the intervals, for the progression A - C#, B♭ - D and B - D# over which the beat frequency suddenly increases to over 20 Hz and then drops back again. These effects contribute to a more interesting aural experience when listening to this temperament compared with the relative blandness of ET. The effects arise because B♭ is offset from ET by -3 cents whereas its major third (D) is not (see Table 2), consequently its detuning contributes to the impression of a distinct key colour in the keys which include it. E flat major, F major, F sharp major and B flat major are examples.
The irregular scatter among the beat frequencies of the major thirds compared to that in ET also endows this temperament with another aspect of key colour which emerges when mutation stops are included in an ensemble, particularly those at 1 3/5 foot pitch such as Tierces or Seventeenths and the equivalent ranks in mixtures. These are tuned pure relative to the unison stops (i.e. so that there is no beat between their fundamental frequencies and the fifth harmonics of the unisons). This results in tuning clashes with the corresponding tempered notes of the scale, and although this also occurs with ET, the effects here depend more strongly on key because of the scattered tuning offsets of the notes shown in Table 2. This is another contribution to key colour which ET does not possess. Nor does any other instrument because none of them has mutation stops, which is another reason why the choice of temperament affects the organ in a unique way as noted earlier.
Together with the slower beats of the fifths tabulated earlier in Table 3, those of the thirds are largely responsible for the colours of this temperament. The slow-beating fifths produce some 'throb' and the faster thirds add 'purr' to an ensemble, at least for 8 and 4 foot stops in the central region of the key compass. These modulations are the underlying phenomena which in fact generate the various key colours. The spectrum of colour within this temperament is not great, and there is certainly nothing comparable to the extreme effects in some other unequal temperaments which render certain keys poor or unuseable. All keys are useable here. It is a mildly unequal temperament which nevertheless possesses greater aural interest than ET itself. However, whether it was designed to be so assumes that my reading of the tuning instructions is correct or at least reasonable.
Although the association with Handel has little foundation, this article has suggested that it is worth revisiting the temperament attributed to him for several reasons. One is the unfortunate fact that many modern realisations of it are wrong because they are incompatible with the complete set of tuning instructions given. These illuminate tuning practices at a time when beat counting was uncommon and therefore, while vague on detail, they nevertheless incorporate an explicit set of overarching constraints which place limits on the possible outcomes. In particular, all fifths must be flat rather than pure or sharpened, and all thirds must be considerably sharp. It is therefore regrettable that the modern realisations of this temperament which are embodied in electronic tuning devices are mostly wrong because they incorporate pure fifths. It is also impossible to conclude, as some authors have done, that the temperament was just another variation on the meantone tunings which were common at that time in Britain. On the contrary, it is difficult to see how the temperament can be other than a mildly unequal one. Consequently it probably lies on the haphazard path which ultimately led to equal temperament in the nineteenth century rather than being merely another example of an already-outmoded meantone approach, although the temperament itself is not equal because the tuning instructions also dictate that the fifths should differ in their deviations from pure.
The frustratingly unfocused tuning instructions nevertheless allow for several subtle variations in realising a version of the temperament, and therefore I took advantage of this to produce my own which has been described here. It addresses the tuning instructions at a level of detail not found elsewhere, providing a reading which I therefore believe to be novel. The outcome is a pleasing well temperament with a range of subtle key colours. Because of the imprecision of the tuning instructions it is not possible to interpret them in a single, unequivocal fashion - the best that can be done is to come up with one of several possible variations on the theme of 'mildly unequal'. That described here has been represented as a set of deviations from equal temperament for each note in the scale, a format which is well suited to most of today's electronic tuning devices and apps, thus it should be reasonably simple to evaluate in practice.
I should like to thank my friend David Hitchin, mathematician, musician and instrument maker, for his knowledgeable perspectives on tuning and temperament during many discussions of the subject.
1. "Twelve Voluntaries and Fugues for the Organ or Harpsichord with rules for tuning by the celebrated Mr
Handel", Book IV, Longman & Broderip, London, c. 1780.
https://tunelab-world.com/tempers/George%20Frederick%20Handel%20well.tem (accessed 28 July 2016)
7. "The rules for 'Through Bass' and for tuning attributed to Handel", Jane Troy Johnson, p. 70, Early Music, February 1989.
8. "The Well-Tempered Organ", Charles A Padgham, Positif Press, Oxford 1986.
9. Padgham's 'Well-Tempered Organ', an article on this website, C E Pykett 2013.
10. Expressing quantities in cents to a precision of three decimal places might seem excessive but in fact it is necessary if substantial rounding or truncation errors are not to accumulate when several arithmetical steps are cascaded. For example, the Pythagorean Comma (-23.46 cents) cannot be usefully represented to a precision of less than two decimals, and this means that intermediate calculations must be performed to a precision of at least three decimals. Spreadsheet calculations are particularly vulnerable to this insidious problem because it is not always obvious that it is happening. It is possible that many of the numerical errors in Padgham's treatise on organ temperaments  arose in this way, though others were clearly a by-product of sloppy typesetting and proof reading.
11. "Unequal Temperaments: Theory, History and Practice", C Di Veroli, Bray Baroque, Ireland, 3rd edition, p. 447, 2013.
12. Quoted by Fred Sturm in .
13. The beat frequencies in the right hand column of Table 3 are approximate values intended only to facilitate discussion rather than for tuning this temperament by ear. One should derive more accurate values expressed (for example) as the number of beats per minute for the latter purpose.